for c
8x
4
−2x
3
−27x
2
+6x+9=0
Let the roots be a,−a,b,c
S
1
=a−a+b+c=−
8
−2
=
4
1
b+c=
4
1
.....(i)
S
3
=−a
2
b−abc+abc−a
2
c=−
8
6
⇒a
2
(b+c)=
4
3
.....(ii)
dividing (i) by (ii)
⇒b+c=
4
1
......(iii)
S
4
=−a
2
bc=
8
9
⇒bc=−
8
3
....(iv)
substituting b from (iii) in (i)
b(
4
1
−b)=−
8
3
4
b−4b
2
=−
8
3
2b−8b
2
=3
8b
2
−2b−3=0
8b
2
+4b−6b−3=0
4b(2b+1)−3(2b+1)=0
(4b−3)(2b+1)=0
⇒b=
4
3
,−
2
1
substituing b in (iv)
⇒c=−
2
1
,
4
3
S
2
=a(−a)+(−a)(b)+bc+ca=
8
−27
⇒−a
2
−ab+bc+ac=
8
−27
The value of s
4
is not satisfied for b=−
2
1
So the roots of the equation are
3
,−
3
,
4
3
,−
2
1
